We define the visibility of the reconstructed image to be \(v=\left(\bar{p}_{s}-\bar{p}_{b}\right)/\left(\bar{p}_{s}+\bar{p}_{b}\right)\) 
in which \(\bar{p}_s\)  and \(\bar{p}_b\)  are the average pixel values of the signal (“IQSE” regions) and background (other regions), respectively. Different from conventional structured illumination techniques in which the information required for image reconstruction is only encoded in the spatial structure of the patterns [1]}, [2]}, [3]}, [4]}, [5]}, the STEP illumination also encodes the information (frequencies) in time. One advantage of introducing the time-domain encoding is that it allows a further improvement of the visibility via halving a successive segment of the data and calculating the cross-spectrum of the two halves. The cross-correlation technique has been demonstrated very effective in weak signal detection [6]}, [7]}, [8]}. Cross-spectrum is the frequency-domain representation of the cross-correlation of two time series signals, which is defined by \(S_{12}\left(f\right)\equiv \mathcal {F}_{\tau }\left\lbrace  g_{1}\star g_{2}\right\rbrace  =\mathcal {F}_{t}\left\lbrace  g_{1}\right\rbrace  \cdot \mathcal {F}_{t}\left\lbrace  g_{2}\right\rbrace \) , where \(\tau \)  is the delay in time, and \(g_{1}\star g_{2}=\sum _{t=0}^{M}g_{1,t}g_{2,t+\tau }\)  is the cross-correlation of two discrete signals \(g_{1}\)  and \(g_{2}\)  with \(M\)  data points. For images reconstructed with a specific \(\beta \) , higher visibility is obtained if the searching of target frequencies is performed in the cross-spectrum of the two halves of the data (Fig.
 REF (b)) rather than in the Fourier spectrum of all the data (Fig. REF (a)). As shown in Fig. REF (b), we divide the whole data set into two segments of the same length, and their cross-correlation is found to be
\(I_{1}\star I_{2}=\frac{M}{2}I_{ij}I_{ij}^{\prime }\sin {2\pi f_{ij}\frac{\tau }{r_{s}}}+\frac{M}{2}\bar{N}_{1}\bar{N}_{2}\) 
